ANALYSIS OF THE DYNAMICS FOR THE SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS DESCRIBING A TUBULAR REACTOR. A Thesis. presented to

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1 ANALYSIS OF THE DYNAMICS FOR THE SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS DESCRIBING A TUBULAR REACTOR A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science by BUSUYI OJO ADEBAYO Prof. David G. Retloff, Thesis Supervisor DECEMBER 017

2 The undersigned, appointed by the dean of the Graduate School have examined the thesis entitled ANALYSIS OF THE DYNAMICS FOR THE SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS DESCRIBING A TUBULAR REACTOR presented by Busuyi Ojo Adebayo, a candidate for the degree of master of science, and hereby certify that, in their opinion, it is worthy of acceptance. Professor David G. Retloff Professor Karl D. Hammond Professor Stephen J. Lombardo

3 ACKNOWLEDGMENTS This is a thesis that was conducted and completed in the period of 015 to 017 at the Department of Chemical Engineering, University of Missouri-Columbia. Foremost, I would like to express my sincere gratitude to my research advisor, Prof. David Retloff, for the support of my M.S. study and research and for his patience, motivation, and enthusiasm with immense and expert knowledge. His guidance throughout my M.S. degree program, research and writing of this thesis cannot be overemphasied. I could not have imagined having a better advisor and mentor for this study. Thank you for believing in me. I would also like to thank my thesis committee members for their invaluable time, encouragement, insightful comments and questions, and also for proof reading this manuscript. And to the Department of Chemical Engineering, I want to say thank you for every dime I was given to fund my program. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii LIST OF FIGURES... v NOMENCLATURE... vi ABSTRACT... viii CHAPTER 1: INTRODUCTION Research Background and Motivation.. 1. Steady-state equations for an adiabatic tubular reactor A priori bounds on C A () and T() Change of variables...10 CHAPTER : METHODOLOGY Transformation of the coupled system Self-adjoint differential and boundary operators Quasimonotonicity of an arbitrary function F = (F 1, F ) Mixed quasimonotonicity of F = (F 1, F ) defined in the BVP Upper and lower solutions for the coupled system Existence of a solution for the coupled system Unique solution and/or multiple solutions to the couple system using Upper and lower solutions method Unique Solution for the coupled BVP using upper and lower solutions Method...33 iii

5 Application of Theorem.1 to the coupled BVP Multiple solutions for the coupled BVP using the upper and lower solutions method Construction of the first pair of upper and lower solutions each for y 1 and y Construction of the second (and disjoint) pair of upper and ` lower solutions each for y 1 and y Results and Discussion Results Discussion.. 47 CHAPTER 3: CONCLUSIONS AND FUTURE WORK Conclusions Future Work...50 BIBLIOGRAPHY...5 iv

6 LIST OF FIGURES Figure Page 1 Schematic view of an adiabatic tubular reactor...4 Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for the pair in Equation (.13) for [0,1] using parameters β 1 = 00, β = Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for the pair in Equation (.13) for [0,1] using parameters β 1 = 150, β = Graphs of y i y i for the pair in Equation (.13) for [0,1] Graph of Equation (.15) Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for the pair in Equation (.16) for [0,1] Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for the pair in Equation (.17) for [0,1] Graphs indicating that the pair in Equation (.16) is disjoint from the pair in Equation (.17) for y 1 for all [0,1] Graphs indicating that the pair in Equation (.16) is disjoint from the pair in Equation (.17) for y for all [0,1] v

7 NOMENCLATURE (y 1, y ) and (y 1, y ) Upper and Lower solutions, respectively (Y 1, Y ) and (Y 1, Y ) Maximal and Minimal solutions, respectively L DD D Length of the reactor Spatial variable Domain of Diffusion Coefficient RR 1 RR Range of (y 1, y ) TR CTR PFR CSTR BVPs ODEs PDEs k k 0 k T C P C A C A0 ( H A ) r A v ρ E Tubular Reactor Continuous Tubular Reactor Plug Flow Reactor Continuous Stirred Tank Reactor Boundary Value Problems Ordinary Differential Equations Partial Differential Equations Reaction rate constant Pre-exponential factor Fluid s thermal conductivity Fluid s specific heat capacity Reactant (A) concentration Reactant (A) concentration before entering the reactor Heat of reaction relative to reactant (A) Rate of decomposition of reactant (A) Fluid s superficial velocity Fluid sdensity Reaction activation energy vi

8 R T T 0 T max Universal gas constant Reaction absolute temperature Reactant temperature before entering the reactor Reaction maximum temperature i Subscript for 1, e.g., x 1, x, y 1, y αα A constant known as the Hölder s exponent δ A number defined as T 0 T max γ The activation energy parameter, defined as E RT max β 1 β Mass Pe clet number Thermal Pe clet number µ 1 The Damko hler number, defined as k 0L µ The quantity defined as ( H AA)k 0 L k T T max N Pr The Prater number, defined as ( µ ) µ 1 D C A0 x 1 The dimensionless concentration, defined as C A C A0 x The dimensionless temperature, defined as T T max a 1, a, b 1, b, k 1 and k The cofficients used to defined the pairs of upper and lower solutions f 1 and f The nonlinear functions of C A and T in the original BVP [Equations (1.4) and (1.b)] F 1 and F The nonlinear functions of, y 1 and y after nondimensionaliation followed by transformation to self-adjoint BVP vii

9 ANALYSIS OF THE DYNAMICS OF THE SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS DESCRIBING A TUBULAR REACTOR Busuyi O. Adebayo Prof. David G. Retloff, Thesis Supervisor ABSTRACT A characteriation of the solution(s) of nonlinear boundary value problems (BVPs) arising from a class of chemical reactions occurring in a tubular reactor is performed. The system in this study is an adiabatic tubular reactor with different mass and thermal Pe clet numbers. Bounds (lower and upper) on the number of solution(s) for the steady state behavior are obtained. The effect of the system parameters Pe clet numbers, Damko hler number, activation energy, etc. on the number of steady-state solution(s) exhibited by the tubular reactor is investigated by applying the mathematical method of upper and lower solutions. The method was first used to show existence of at least one solution to the BVP for two sets of parameters. Uniqueness of the solution was also proved using this method combined with a theorem, Theorem.1. Based on restrictions on the parameters of the BVP, the existence of multiple solutions was proved as well. Results show that for large Pe clet numbers and activation energy, and for sufficiently small Damko hler number and reactor length, the solution to the boundary value problem is unique. For small Pe clet numbers and activation energy, and for large Damko hler number and reactor length, there exist at least three solutions for the BVP. The conclusion of all this is that the adiabatic tubular reactor is an intermediate model between the adiabatic plug flow reactor model and the adiabatic continuous stirred tank reactor model. viii

10 CHAPTER 1: INTRODUCTION The tubular reactor (TR) model is used to describe chemical reactions in continuous, flowing systems of cylindrical geometry. This is the reason why the TR is sometimes called the continuous tubular reactor (CTR). In a TR, the feed typically enters at one end of a cylindrical tube and the product stream leaves at the other end. The long tube and the lack of provision for stirring prevent complete mixing of the fluid in the tube. Hence, the properties of the flowing stream will vary from one point to another, namely in both radial and axial directions. The tubular reactor considered in this research is the non-ideal plug flow reactor (PFR); that is, no radial variation of properties, with the non-ideality from axial dispersion. The dynamics of a tubular reactor give rise to one or more differential equations (partial differential equations (PDEs) if operating in an unsteady/transient state or ordinary differential equations (ODEs) if operating at a steady state). For nonisothermal tubular reactors in particular, the dynamics are described by coupled nonlinear PDEs or ODEs. The main sources of nonlinearities in the dynamics are concentrated in the kinetic terms of the constitutive equations. The existence of Arrhenius-type nonlinearities in the kinetics can generate multiple steady states, either stable or unstable, and in practical applications, the unstable steady states may correspond to the operating points of interest [1,]. Our task is to find solution(s) to the equations and discuss the existence, uniqueness, and/or multiplicity of the solution(s) of the equations. Recent 1

11 experimental, numerical, and theoretical results show that an adiabatic tubular reactor, in which occurs a simple first-order irreversible exothermic chemical reaction, can exhibit multiple steady-states. The study of the steady-state multiplicity and stability was the object of intense research in the 1960s and 1970s: most of the pertinent chemical engineering literature and results are reviewed by Varma and Aris [3], by Aris [4], and by Luss [5]. Multiple steady states have been observed experimentally, for example, in adiabatic tubular reactors [6,7]. However, a numerical method will work only if the solution of the problem exists [8]. With this in mind, many authors devoted their time to studying the existence of solutions. 1.1 RESEARCH BACKGROUND AND MOTIVATION Many researchers (in the fields of chemical engineering, mathematics, and others) have worked extensively on the solution(s) to the nonadiabatic tubular reactor with both identical and differing mass and thermal Pe clet numbers and the adiabatic tubular reactor with identical mass and thermal Pe clet numbers [9 1]. However, very little has been done on or is known about the uniqueness and/or multiplicity of solution(s) to the adiabatic tubular reactor when the mass and thermal Pe clet numbers are different. For the nonadiabatic tubular reactor when the mass and thermal Pe clet numbers are equal, tremendous achievements have been made: Donald S. Cohen and Theodore W. Laetsch [11], D. Luss and N.R. Amundson [5], D. Dochain [1], etc. have shown that multiple steady-states exist if the diffusion coefficient is sufficiently large,

12 while for the nonadiabatic tubular reactor when the mass and thermal Pe clet numbers are different, research by Retloff et al [1], Laabissi et al [], and A. Varma & R. Aris [3] have shown that multiple steady-state solutions are possible for various values of the system parameters values, e.g. the activation energy. For the adiabatic tubular reactor when the mass and thermal Pe clet numbers are equal, D. Cohen and T.W. Laetsch [11] has shown that multiple solutions can exist for certain values of the system parameters, especially when the activation is large. In fact, for this case, the original coupled boundary value problem (BVP) easily reduces to a scalar BVP, which is easily analyed. However, for the case in this work, that is, the adiabatic tubular reactor when the mass and thermal Pe clet numbers are different, the existence, uniqueness and/or multiplicity remains of solutions to the BVP remains an open question. Here, the mass and energy balances cannot be decoupled, making the analysis more difficult. Determining the existence, uniqueness, and/or multiplicity of solution(s) when the two Pe clet numbers differ for specific parameter sets is the goal of this work. 1. STEADY-STATE EQUATIONS FOR AN ADIABATICALLY OPERATED TUBULAR REACTOR The tubular reactor, as considered in this research, is defined as a non-ideal PFR; that is, no radial variation of properties, with the non-ideality being from axial dispersion; thus, the name axial-dispersion model tubular reactor. A schematic of an adiabatic tubular reactor is shown in Figure 1. 3

13 Figure 1: Schematic of an adiabatic tubular reactor. The reactant, A, forms the product, B, in an isomeriation. In this research, we considered the case of a single irreversible, first order, exothermic isomeriation reaction in an adiabatic tubular reactor. The reaction is A B, And the rate of consumption of A is r A = kc A, (1.1) where k = k 0 exp E RT. The dynamics of the system are derived from material and energy balance considerations. The steady-state dynamics of an adiabatic tubular reactor with 4

14 axially-dispersed plug flow for the simple irreversible exothermic reaction described by Equation (1.1) result in the following set of ODEs: D d C A d v dc A d = r A (1.a) and k T d C A d vρc dc A p d = ( H A)( r A ), (1.b) The boundary conditions are B. C. 1: D dc A d (0) + vc A(0) = vc A0 k T dt d (0) + vρc pt(0) = vρc p T 0 (1.3a) (1.3b) B. C. : dc A d (L) = 0 (1.3c) dt d (L) = 0 (1.3d) Note that C A0 is the concentration of A before the inlet of the reactor and T 0 is the inlet temperature just before the reaction starts, and they are assumed to be given/known. In the above equations, is the spatial variable in meter [m]; L [m] is the length of the reactor, where 0 L; v is the fluid s superficial velocity [m s 1 ]; T > 0 is the process component temperature in Kelvins [K]; C A > 0 is the reactant A concentration [kg m 3 ]; ( H A ) is the heat of reaction with respect to A [kj kg 1 ], 5

15 are where ( H A ) < 0 for exothermic reactions, and > 0 for endothermic reactions; ρ is the fluid s density [kg m 3 ]; and C p is the specific heat capacity [kj kg 1 K 1 ]. Note: (1) Diffusion terms (i.e., dispersion) in Equations (1.a) are based on the Fick s second law and the heat diffusion equation, but they also include contributions from fluid flow in the axial direction [13]. () In Equations (1.), the reaction/kinetic term corresponds to the kinetics of a non-isothermal reaction with first order kinetics with respect to the reactant concentration C A and Arrhenius-type dependence with respect to the temperature T. The term indeed closely couples the mass and energy balance equations. (3) The boundary conditions in Equations (1.3) are known as the Danckwerts conditions [14]. In deriving/formulating these equations, the following assumptions have been implicitly made: (1) The fluid thermal (C P ) and physical parameters (ρ, k T, D)0T independent of temperature and composition. () The heat of reaction, that is, the reaction enthalpy change ( H A ), is independent of temperature. (3) Radial gradients of the temperature and concentration are negligible. (4) The fluid spatial velocity is constant in the axial direction. 6

16 (5) Kinetic and potential energy changes are negligible with respect to internal energy changes. (6) There is no shaft work involved, that is, the only work is flow work. (7) The dispersive fluxes of mass and heat can be described through Fick s and Fourier s law, respectively, with effective/overall mass and heat diffusion coefficients. We next establish specific upper and lower bounds for the dependent variables C A and T by assuming that they are solutions of Equations (1.) and (1.3) and show that C A is a decreasing function of while (for the exothermic case) T is an increasing function of. The upper bounds will identify the scaling factors used to obtain dimensionless dependent variables. 1.3 A PRIORI BOUNDS ON CC AA () AND TT() Lemma 3.1 C A () is a decreasing function of, and the maximum value of C A () is C A0. Proof First, observe that r A is negative for C A () > 0 and rewrite Equation (1.a) as: d C A d v dc A D d = r A D = f 1 0 (1.4) Integrating the inequality in Equation (1.4) with respect to from 0 to L and using the boundary conditions [Equations (1.3)] yields v D C A(L) + v D C A 0 0, 7

17 from which it follows that C A0 C A (L). (1.5) As C A is (0, L), that is, C A is twice continuously differentiable on (0, L), there exists a segment of (0, L) for which dc A d 0. Assuming C A(0) > C A0, then dc A (0) > 0. Thus d there exists 1 (0, L) where dc A d ( 1) = 0 and C A ( ) is a maximum. Therefore, d C A ( d 1) < 0. This a contradiction: it contradicts Equation (1.4) leading to the conclusion that dc A d (0) < 0. In addition, C A() is a monotonically decreasing function of, otherwise it would achieve a maximum value in (0, L) and result in a contradiction of Equation (1.4). Hence, C A () < C A0. Lemma 3. For an exothermic reaction, T() is an increasing function of, and the maximum and minimum values of T() are T(L) and T 0 > 0, respectively. Proof We observe that ( H A )( r A ) is negative for an exothermic forward reaction and repeat Equation (1.b) here for convenience: k T d C A d vρc dc A p d = ( H A)( r A ) = f 0. (1.b) Integrating the inequality [Equation (1.b)] with respect to from 0 to L and using the boundary conditions [Equations (1.3)] yields vρc PT(L) k T + vρc pt 0 k T 0, (1.6) from which it follows that: 8

18 T 0 T(L). (1.7) Now assume T(0) < T 0 which yields dt (0) < 0. It follows that there exists a d 1 (0, L) where dt ( d 3) = 0, T(L) is a minimum at 1, and d T d > 0. This contradicts Equation (1.b). Hence, T 0 T(0). It follows that T() is an increasing function of in the interval [0, L], otherwise it contradicts Equation (1.b). Hence, T 0 is the minimum temperature. To determine the maximum temperature, T max, combine Equation (1.4) and Equation (1.b) to obtain: d T d vρc p dt k T d = ( H A)D k T d C A d v D dc A. (1.8) d Integrating Equation (1.8) from = 0 to = L yields T(L) = T 0 ( H A) ρc p C A0 C A (L). (1.9) Therefore, defining T max as T max = T 0 ( H AA) ρc p C A0, (1.10) then T(L) T max. 9

19 1.4 CHANGE OF VARIABLES We now do a change of variables, (C A, T) (x 1, x ) so that (x 1, x ) [0,1] [δ, 1]. To accomplish this, let x 1 = C A, x C = T,, δ = T 0, γ = E, β A 0 T max L T max RT 1 = vl, β max D = vρc pl, k T µ 1 = k 0L D k 0L v vl D, and µ = ( H AA)k 0 L k T T max C A0, where C A0 is the initial concentration, T max is the reference temperature, k T ρc p is the thermal diffusion coefficient [m s 1 ], and D is the mass diffusion coefficient [m s 1 ]. Note that β 1 and β are the mass and thermal Pe clet numbers, respectively, and µ 1 is the Damko hler number. The quantity ( µ ) µ 1 = ( H A)k 0 L C A0 k 0 L k T T max D = ( H A )D = N T Pr max k T C A0 is the maximum fractional temperature difference or simply the Prater number (N Pr ), where ( H A )D k T is the maximum temperature difference. It was reported by Boglaev [15] that the Damko hler number, µ 1, can vary over a wide range, while the Prater number, N Pr varies in the range 0.01 N Pr 1. Therefore, we choose the restriction that 0.01 N Pr ( µ ) µ 1 1. (1.11) With this change of variables, Equations (1.) become 10

20 d x 1 d β dx 1 1 d = µ 1exp γ x x 1 (1.1a) and d x d β dx d = ( µ )exp γ x x 1 (1.1b) for DD = (0,1). Using the same scaling, Equations (1.3) become dx 1 d (0) + β 1x 1 (0) = β 1 (1.13a) dx d (0) + β x (0) = β δ (1.13b) dx 1 d (1) = 0 (1.13c) dx d (1) = 0 (1.13d) Note: D is the domain of the existence of solutions of the elliptic boundary value problem (BVP) described by Equations (1.1) and (1.13) and let denotes the boundary of D and DD the closure of DD. Since DD = (0,1), then = 0 and 1 and DD DD = [0,1]. The a-priori bounds for the steady-state reactant concentration and temperature are obvious from the system chemistry and physical arguments: x 1 is non-negative with 0 x 1 () < 1, while x is positive with 0 < δ < x () 1, where [0,1]. 11

21 REMARK 1.1 Since δ < x 1, it must be that the following equation holds: where: min temperature + max fractional temperature rise max temperature (1.14) minimum (min) dimensionless temperature = δ maximum (max) fractional temperature rise = N Pr ( µ ) µ 1 maximum (max) dimensionless temperature=1 Hence, we have that δ + N Pr 1 δ + ( µ ) µ 1 1 Therefore, knowing/choosing δ, we get ( µ ) µ 1 1 δ. Therefore, we choose the restriction that ( µ ) µ 1 1 δ. (1.15) Note: Equation (1.15) is a refinement of Equation (1.11). And this restriction will be enforced in Chapter where the upper and lower solutions conditions for the BVP are proven. 1

22 CHAPTER : METHODOLOGY The mathematical method adopted in this research is the method of upper and lower solutions together with its associated monotone sequence iteration. The first steps in the method of upper and lower solutions have been given by Picard in 1890 [16] for PDEs and three years later [17] for ODEs. In both cases, the existence of a solution is guaranteed from a monotone iterative technique. The existence of solutions for Cauchy equations was proved by Perron in 1915 [18]. In 197, Müller extended Perron's results to initial value systems in [19]. In 1931, Dragoni [0] introduced the notion of the method of upper and lower solutions for ODEs with Dirichlet boundary value conditions. It is well known that the upper and lower solutions method is a powerful tool for proving the existence of a solution to a boundary value problem. It has been used to deal with many two-point and multipoint boundary value problem ODEs (see, e.g., [1 3] and references therein). Many people pay attention to the existence, uniqueness, and/or multiplicity of solutions or positive solutions for boundary value problems of nonlinear differential equations by means of some fixed-point theorems, such as the Krasnosel'skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the Schauder fixed-point theorem [4 8]. Before using the method of upper and lower solutions to discuss and prove the existence, uniqueness, and/or multiplicity of solution(s), let us first transform the non-self-adjoint BVP to a corresponding self-adjoint BVP and also discuss the method 13

23 of upper and lower solutions because these, as shown by Pao [9], are critical tools in order to establish the uniqueness and/or multiplicity of solutions of the BVP..1 TRANSFORMATION OF THE COUPLED SYSTEM Let us transform the system using the following equations: x 1 = 1 e β 1 y 1 (.1a) x = δ + e β y (.1b) Substituting Equations (.1) into the two Equations (1.) and the boundary conditions [Equations (1.3)] gives d y 1 d = β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y F 1 (, y 1, y ) (.a) d y d = β y + ( µ )e β 1 e β 1 y 1 e β δ+e y F (, y 1, y ) (.b) in DD, and the transformed boundary conditions dy 1 d (0) + β 1 y 1(0) = 0 = h 1 (.3a) dy 1 d (1) + β 1 y 1(1) = 0 = h (.3b) 14

24 dy d (0) + β y (0) = 0 = h 3 (.3c) dy d (1) + β y (1) = 0 = h 4. (.3d) The transformed BVP operators are now LL i d d, and BB i d + β i at the boundary = 0, and d + β i at the boundary = 1. d d The transformed system with differential operators (LL i ) and boundary operators (BB i ) forms a self-adjoint system as shown in the following section..1.1 SELF-ADJOINT DIFFERENTIAL AND BOUNDARY OPERATORS Given an arbitrary operator LL, then the adjoint of LL is an operator denoted here by LL such that v, LLu = u, LL v (.4a) where.,. represents the inner product defined by: AA v, LLu v LLu ddd DD (.4b) for real-valued functions u and v. If LL = LL, then the operator LL is said to be self-adjoint. For the case of the transformed differential and boundary operators LL i and BB i above, we have 15

25 AA v, LLu v LLu ddd = v d u d d = v d d du d d DD = v du 1 d 0 1 du d 0 dv v du 1 d 0 1 du dv d d 0 1 du d v d 0 1 dv d 0 du d d = v du d u dv 1 d u d v d d 0 From the boundary conditions [Equations (1.3)], the expression v du d u dv 1 d = 0. 0 Hence, we have 1 1 v, LLu v d u d d = u d v d d u, LL v (.5) 0 0 Therefore, we conclude that the differential operators LL i in Equations (.) together with the boundary conditions operator BB i in Equations (.3) form a self-adjoint system. Other features brought about by this choice of transformation are: (1) preservation of the positivity of the solutions, () homogeneity of the boundary conditions, which results from the selfadjointness of the operators. Note: Self-adjointness of the differential and boundary condtions operators is a requirement to apply Theorem.1 in section...1 to the BVP, thus the reason for the transformation of our non-self-adjoint BVP to self-adjoint BVP. 16

26 .1. QUASIMONOTONICITY OF AN ARBITRARY FUNCTION FF = (FF 11, FF ) Definition.1 A function F = (F 1, F ) is called quasimonotonically nondecreasing (resp., nonincreasing) in RR 1 RR if both F 1 and F are quasimonotonically nondecreasing (resp., nonincreasing) for (y 1, y ) RR 1 RR. When F 1 is quasimonotonically nondecreasing and F is quasimonotonically nonincreasing (or vice versa), then F is called mixed quasimonotone, where RR 1 RR is the range of (y 1, y ). Note: The prefix quasi as used here means that the function F = (F 1, F ) or F i is a multivariable function, that is, at least two independent variables. Hence, if we know the monotonicity with respect to one variable, then the monotonicity with respect to other variables is left open. The function F is said to be quasimonotone in RR 1 RR if it has any one of the quasimonotone properties in Definition.1. As usual, we call F a 1 -function in RR 1 RR if both F 1 and F are continuously differentiable in (y 1, y ) for all functions (y 1, y ) RR 1 RR. F is called a quasi 1 -function in RR 1 RR if F 1 is continuously differentiable in y and F is continuously differentiable in y 1 for all functions (y 1, y ) RR 1 RR. It is clear that every 1 -function is a quasi 1 -function, but the converse is not necessarily true. Hence, if F is a 1 -function or a quasi 1 -function then three types of quasimonotone functions in Definition.1 are reduced to the form F 1 y 0, F y 1 0 (.6a) 17

27 F 1 y 0, F y 1 0 (.6b) F 1 y 0, F y 1 0 (.6c) for (y 1, y ) RR 1 RR MIXED QUASIMONOTONICITY OF FF = (FF 11, FF ) DEFINED IN THE BVP For the function F 1 defined in Equation (.a), that is, F 1 (, y 1, y ) = β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y, we have, F 1 (, y 1, y ) y = γγµ 1 e (β1 β) δ + e β y 1 e β 1 y 1 e β δ+e y for (, y 1, y ) DD yy, yy, since γγ and µ 1 are positive constants. To prove that F 1 (, y 1, y ) y 0, (.7a) it suffices to show that 18

28 1 e β 1 y 1 0 for (, y 1, y ) DD yy, yy, (.7b) where DD yy, yy is the domain of (, y 1, y ); yy = (y 1, y ) and yy = (y 1, y ) are the upper and lower solutions vector respectively, that will define in the next section. One way to prove Equation (.7b) is that if 1 e β 1 y 1 0 and 1 e β 1 y 1 0, then we have that 1 e β 1 y 1 0 for (, y 1, y ) DD yy, yy. However, because Equation (.7b) depends on our choice of y 1 and y 1 (upper and lower solutions to the BVP), which are not yet available, an alternative is to consider the bounds on x 1 and then the corresponding transformation to y 1. We note that since 1 e β 1 y 1 = x 1 : 0 x 1 < 1, then we have that 1 e β 1 y 1 0 for (, y 1, y ) DD yy, yy. For the function F defined in Equation (.b), that is, F (, y 1, y ) = β y + ( µ )e β 1 e β 1 y 1 e β δ+e y, we have F (, y 1, y ) y 1 = ( µ )e β1 β e β δ+e y. Clearly, F (, y 1, y ) y 1 < 0 (.8) 19

29 for all (, y 1, y ) DD yy, yy, since ( µ ) is a positive constant for our exothermic isomeriation. Thus, we conclude that F 1 (, y 1, y ) is quasimonotonically nondecreasing and F (, y 1, y ) quasimonotonically nonincreasing, and thus complete the proof that the function F = (F 1, F ) is mixed quasimonotone. Note: Knowing the type of quasimonotonicity the function F = (F 1, F ) is allows us to define an accurate iterative scheme for the solution to the BVP. For the case when the function F = (F 1, F ) is either quasimonotonically nondecreasing or nonincreasing, the iterative scheme normally converges to the BVP true solutions depending on the choice of upper and lower solutions chosen to start the iteration with. But for the mixed quasimonotonocity, the iterative scheme either converges to the true solution or just gives us the closest region where the true solution lies. Then, for this case of mixed quasimonotonicity, there is a mathematical condition to check to know if the iteration actually converges to the true solution or otherwise. Moreover, the iterative scheme for each type of quasimonotonicity defers in one way or another [9].. UPPER AND LOWER SOLUTIONS FOR THE COUPLED SYSTEM The method of upper and lower solutions is a useful mathematical tool that one uses when trying to prove the existence of a solution to a differential equation or system of differential equations [9 3]. In fact, it can also be used to prove multiplicity of solutions, provided there exist more than one disjoint pair of upper and lower 0

30 solutions [9]. First, we define upper and lower solutions for the coupled system, give one or two theorems about the properties pertaining to them, and then apply them to the BVP of interest, specifically the transformed one [Equations (.) and (.3)]. Definition (.) Suppose that for the mixed quasimonotonic function F = (F 1, F ): F 1 quasimonotonically nondecreasing and F quasimonotonically nonincreasing, there exists a pair of functions yy = (y 1, y ) and yy = (y 1, y ) which satisfies the boundary inequalities dy 1 d (0) + β 1 y 1(0) 0 (.9a) dy 1 d (1) + β 1 y 1(1) 0 (.9b) dy 1 d (0) + β 1 y 1(0) 0 (.9c) dy 1 d (1) + β 1 y 1(1) 0 (.9d) dy d (0) + β y (0) 0 (.9e) dy d (1) + β y (1) 0 (.9f) dy d (0) + β y (0) 0 (.9g) dy d (1) + β y (1) 0 (.9h) The pair of functions yy = (y 1, y ) and yy = (y 1, y ) αα DD (DD) are called ordered upper and lower solutions of Equations (.) and (.3) if they satisfy the following conditions: (i) The boundary inequalities [Equations (.9)], (ii) LL 1 y 1 + F 1 (, y 1, y ) 0 LL 1 y 1 + F 1 (, y 1, y ), (.10) (iii) LL y + F (, y 1, y ) 0 LL y + F (, y 1, y ), (.11) 1

31 (iv) yy and yy are ordered, that is, yy yy, where yy = (y 1, y ) and yy = (y 1, y ). This is equivalent to y 1 y 1 and y y. (.1) DD is the closure of DD. Remark.1 By yy and yy αα DD, we mean yy and yy are αα Ho lder continuous in DD, and by yy and yy (DD), we mean yy and yy are at least twice continuously differentiable in DD. Thus, yy and yy αα DD (DD) meant yy and yy are both αα Ho lder continuous in DD and at least twice continuously differentiable in DD. And the number αα is called the Hölder s exponent. In summary, the method of upper and lower solutions allows us to ensure the existence of a solution of the considered problem lying between the lower and the upper solutions, that is, we have information about the existence and location of the solutions. So, the problem of finding a solution of the considered problem is replaced by that of finding two well-ordered functions that satisfy some suitable inequalities. Following these results, there have been a large number of works in which the method has been developed for different kinds of boundary value problems; first-, second- and higher-order ordinary differential equations with different types of boundary conditions such as, among others, periodic, mixed, Dirichlet, or Neumann conditions, have been considered [3]. Also, PDEs of first and second-order, have been treated in the literature [33 35].

32 ..1 EXISTENCE OF A SOLUTION FOR THE COUPLED SYSTEM The results in this section will guarantee the existence of at least one solution to the coupled BVP. Here, existence of a solution (at least one) will be proven by construction of a pair of upper and lower solutions for the transformed BVP, and its dependence on the system parameters. Though there is no hard rule on how to construct upper and lower solutions for ODEs or PDEs, a brief guide on how to construct for ODEs is discussed by Tam [36]. The proposed pair of upper and lower solutions is as shown below. y 1 = k 1 e a β1 1 y 1 = b 1 π sin (π) y = k e a β y = b sin (π) (.13) π This pair was defined so that it covers any/some part (i.e., not necessarily the entire solution range) of the solution range for all [0,1]. Every solution range is bounded by upper and lower solutions. Based on the two parameters values sets (i) β 1 = 100, β / = 75 and (ii) β 1 = 75, β = 50 together with µ 1 = 0.35, µ = 0.1, δ = 0.005, γγ = 1.5, and the constants a 1 = a = 0.5, b 1 = b = , k 1 = 0.5, k = 0.5 δ, the upper and lower solution conditions (i-iv) [Equations (.9-.1)] are proved as follows. Proof of condition i [Equations (.9)]: From Equation (.9a), 3

33 dy 1 d (0) + β 1 y 1(0) 0, we have d k 1 e a β 1 1 d (0) + β 1 k 1e a β 1 1 k 1 β 1 (0) = a 1 + k 1β 1 = k 1β 1 (1 + a 1) 0, 1 + a 1 0 a 1 1. Hence, we have dy 1 d (0) + β 1 y 1(0) 0 for a 1 1. From Equation (.9b), dy 1 d (1) + β 1 y 1(1) 0, we have d k 1 e a 1 β 1 d (1) + β 1 k 1e a β 1 1 k 1 β 1 β 1 (1) = a 1 e a 1 + k 1β 1 β 1 e a 1 = k 1β 1 β 1 e a 1 (1 a1 ) 0, 1 a 1 0 a 1 1 a 1 1. Hence, we have dy 1 d (1) + β 1 y 1(1) 0 for a 1 1. Therefore, we conclude that for Equations (.9a) and (.9b) to hold simultaneously, 1 a 1 1. From Equation (.9c), 4

34 dy 1 d (0) + β 1 y 1(0) 0, we have d b 1 π sin(π) d (0) + β 1 b 1 π sin(π) (0) = b 1 π cos(0) + β 1 b 1 π sin(0) = b 1 π. Clearly, dy 1 d (0) + β 1 y 1(0) = b 1 π < 0 b 1 < 0 b 1 > 0. This implies that Equation (.9c) holds for b 1 > 0. From Equation (.9d), dy 1 d (1) + β 1 y 1(1) 0, we have d b 1 π sin (π) d (1) + β 1 b 1 π sin(π) (1) = b 1 π cos(π) + β 1 b 1 π sin(π) = b 1 π. Clearly, dy 1 d (1) + β 1 y 1(1) = b 1 π < 0 b 1 < 0 b 1 > 0. This implies that Equation (.9d) holds for b 1 > 0. For Equation (.9e), dy d (0) + β y (0) 0, we have d k e a β d (0) + β k e a β k β (0) = a + k β = k β (1 + a ) a 0 a 1 Hence, we have dy d (0) + β y (0) 0 5

35 For Equation (.9f), dy d (1) + β y (1) 0, we have d k e a β d (1) + β k e a β k β β (1) = a e a + k β β e a = k β β e a (1 a ) 0 1 a 0 a 1 a 1. Therefore, we conclude that for Equations (.9e) and (.9f) to hold simultaneously, 1 a 1 1. From Equations (.9e) and (.9f) above, we have that 1 a 1, but likewise, let us restrict to 1 < a < 1. Hence, we have: dy d (1) + β y (1) 0 From Equation (.9g), dy d (0) + β y (0) 0, we have d b π sin (π) d (0) + β b π sin (π)(0) = b π cos(0) + β b π sin(0) = b π In fact, dy d (0) + β y (0) = b π < 0 b < 0 b > 0 From Equation (.9h), 6

36 dy d (1) + β y (1) 0, we have d b π sin (π) d (1) + β b π sin (π)(1) = b π cos(π) + β b π sin(π) = b π Likewise, dy d (1) + β y (1) = b π < 0 b < 0 b > 0 Proof of condition ii [Equation (.10)]: LL 1 y 1 + F 1 (, y 1, y ) 0 LL 1 y 1 + F 1 (, y 1, y ) The left hand side (L. H. S. ) inequality: LL 1 y 1 + F 1 (, y 1, y ) 0 L. H. S. = d y 1 d β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y = d k 1 e a 1 β 1 d β 1 k 1 e a 1 β 1 +µ 1 e β 1 1 e β 1 k 1 e a β 1 1 e β δ+e k e a β = β 1 k 1 e a 1 β 1 (1 a 1 ) +µ 1 e β 1 1 k 1 e (1 a 1 )β 1 e δ+k e (1 a )β For values of µ 1, we need 1 a 1 = (1 a 1 )(1 + a 1 ) 0 for the L. H. S. to be 0. 7

37 An easy way to show that the L. H. S. 0 for all values of [0,1] is the graphical method; this is shown in Figure for β 1 = 100 and β / = 75 and Figure 3 for β 1 = 75 and β / = 50. Note: We can refine the limit on a i as 1 < a i < 1, instead of 1 a i 1 to avoid any possibility of trivial solution. For the first parameter set where β 1 = 00 and β = 150, we have Figure. Figure : Graphs of conditions (ii) and (iii) for Equation (.13) for [0,1] using parameters β 1 = 00, β 1 = 150, µ 1 = 0.35, µ = 0.1, δ = 0.005, γγ = 1.5; and constants a 1 = a = 0.5, b 1 = b = , k 1 = 0.5, k = 0.5 δ. And for the other parameter set where β 1 = 150 and β = 100, we have Figure 3. 8

38 Figure 3: Graphs of conditions (ii) and (iii) for Equation (.13) for [0,1] using parameters β 1 = 75, β = 50, µ 1 = 0.35, µ = 0.1, δ = 0.005, γγ = 1.5; and constants a 1 = a = 0.5, b 1 = b = , k 1 = 0.5, k = 0.5 δ. The right hand side inequality, 0 LL 1 y 1 + F 1 (, y 1, y ), is equivalent to LL 1 y 1 + F 1 (, y 1, y ) 0 L. H. S. = d y 1 d β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y d b 1 = π sin (π) d β 1 b 1 π sin(π) +µ 1 e β 1 1 e β 1 b 1 sin (π) e π β δ+e b πsin (π) 9

39 = b β 1 π sin(π) +µ 1 e β 1 1 b 1 π eβ1 sin (π) e δ+ b β π e sin (π) This is 0 as shown in Figures and 3. Proof of condition iii [Equation (.11)]: LL y + F (, y 1, y ) 0 LL y + F (, y 1, y ) The left hand side (L.H.S.) inequality: LL y + F (, y 1, y ) 0 L. H. S. = d y d β y + ( µ )e β 1 e β 1 y 1 e β δ+e y = d k e a β d β e a β +( µ )e β 1 e β 1 b 1 π sin (π) e β δ+e k e a β L. H. S. = β k e a β (1 a ) +( µ )e β 1 b 1 π eβ1 sin (π) e δ+k e (1 a )β For the chosen parameter values, this is 0 as shown in Figures and 3. 30

40 The right hand side inequality: 0 LL y + F (, y 1, y ); this is equivalent to LL y + F (, y 1, y ) 0 L. H. S. = d y d β y + ( µ )e β 1 e β 1 y 1 e β δ+e y d b = π sin (π) d β b π sin(π) +( µ )e β 1 e β 1 k 1 e a β 1 1 e β δ+e b π sin(π) L. H. S. = b 1 + β π sin(π) +( µ )e β 1 k 1 e (1 a 1 )β 1 e δ+ b β π e sin(π) For the indicated choice of parameters, this is 0 as shown in Figures and 3. Proof of condition iv [Equation (.1)]: yy and yy are ordered, that is, yy yy, where yy = (y 1, y ) and yy = y 1, y. This implies that y 1 y 1 and y y. This is shown in Figure 4. 31

41 Figure 4: Graph of y i y i for the pair in Equation (.13) for [0,1] using β 1 = 00, β = 150, a 1 = a = 0.5, b 1 = b = , k 1 = 0.5, k = 0.5 δ and δ = UNIQUE SOLUTION AND/OR MULTIPLE SOLUTIONS TO THE COUPLED BVP USING THE UPPER AND LOWER SOLUTIONS METHOD As a direct consequence of the transformed Danckwerts BCs [Equations (.3)], the TR model reduces to the continous stirred tank reactor (CSTR) and the plug flow reactor (PFR) models for very large and very small axial dispersion coefficients (equivalent to very small and very large Pe clet numbers, β i, i = 1,), respectively. The multiplicity of solutions to the CSTR has been well established, at least for simple reacting systems [1,43,44]. On the other hand, the PFR model does not admit any multiple steady states, since it is constituted by initial-value first-order ordinary differential equations (ODEs) [43,44]. 3

42 ...1 UNIQUE SOLUTION FOR THE COUPLED BVP USING UPPER AND LOWER SOLUTIONS METHOD Since only a positive real solution is of interest here, and if the maximal (Y 1, Y ) and the minimal (Y 1, Y ) solutions are strictly positive, it is possible to impose a condition in order to ensure the uniqueness of a positive solution [9]. The existence and uniqueness of this positive solution is ensured if there exists a pair of ordered nonnegative upper and lower solutions yy = (y 1, y ), yy = (y 1, y ) with y i 0, that is, y i not identically ero for all [0,1], where i = 1,. In Theorem.1, we show that if F = (F 1, F ) satisfies either one of the conditions F 1(, y 1, y ) ; (, y y 1 y 1, y ) DD yy, yy > 0, h 1 = 0, h 0 (.14a) 1 or F (, y 1, y ) ; (, y y y 1, y ) DD yy, yy > 0, h 1 0, h = 0 (.14b) then for the mixed quasimonotone functions the BVP has a unique positive solution in the sector yy, yy. Theorem.1 Let yy = (y 1, y ), yy = (y 1, y ) in αα DD (DD) be the ordered nonnegative upper and lower solutions of the given BVP with y i 0, i = 1,; and let LL i, BB i be self-adjoint. Assume that (F 1, F ) is mixed quasimonotone and either 33

43 F 1 (, y 1, y ) > 0 and Equation (.14a) holds y or F 1 (, y 1, y ) < 0 and Equation (.14b)holds. y Then (Y 1, Y ) = (Y 1, Y ) and is the unique positive solution of the BVP in yy, yy. Proof The proof of Theorem.1 can be found in Pao [9]. Remark.1 Theorem.1 is a slightly modified version of Theorem in Pao [9] in that the mixed quasimonotonicity is swapped. However, for F = (F 1, F ), the proof remains the same APPLICATION OF THEOREM.1 TO THE COUPLED BVP For unique solution of the BVP, the same pair of functions [Equation(.13)] as used to prove the existence of a solution is proposed for an upper and lower solution pair, y 1 and y, but with the following set of restrictions on parameters: x 1 (0) k 1 < 1, k 1 δ, a i are constants whose limits are to be determined to satisfy the upper and lower solutions conditions [Equations (.9-.1)], and b i 0 +, i.e., the b i parameters are vanishingly small and positive. We chose the two Peclet numbers, β 1 = 150, β = 10; the Damkoler number, µ 1 = 0.35; µ = 0.1, δ = 0.005, γγ = 1.5; and the constants a 1 = a = 0.5, b 1 = b = , k 1 = 0.975, k = 1 δ considered for establishing uniqueness. 34

44 Remark.: The proposed choice of the pair of upper and lower solutions here is such that it covers the entire solution range to be sure of uniqueness. Note: This choice of the parameters of the upper and lower functions/solutions was made so that the entirety of the solution range of the BVP is covered by considering the upper and lower bounds on the varables y 1 and y. This was actually from my personal intuition. Next, we prove the other hypotheses in the uniqueness Theorem.1. (i) F 1 (, y 1, y ) = β 1 y 1 + µ 1 e β 1 1 e β 1 y 1 e β δ+e y Therefore, F 1 (, y 1, y ) = β 1 y 1 + µ 1 e β 1 1 e β 1 e y 1 β δ+e y F 1(, y 1, y ) = µ 1 y 1 y 1 y e 1 β δ+e y < 0 Clearly, this does not satisfy Equation (.a). (ii) F (, y 1, y ) = β y + ( µ )e β 1 e β 1 y 1 e β δ+e y 35

45 Therefore, F (, y 1, y ) y = β + ( µ )e β 1 eβ1 y y 1 β e δ+e y F (, y 1, y ) 1 = ( µ y y )e β eβ1 y 1 yy γγe β y δ + e β y 1 γ β e δ+e y With h 1 = 0, h = 0, and 1 e β 1 y 1 > 0, then Equation (.14b) holds, that is, F (, y 1, y ) 1 = ( µ y )e β eβ1 y 1 γγe β y δ + e β y 1 yy y = ( µ )e β 1 eβ1 yy y 1 γ β e δ+e y γγe β y δ + e β y 1 γ β e δ+e y > 0 > 0 if and only if γγe β y δ + e β y 1 > 0. By recalling the substitution x = δ + e β y, then γγe β y δ + e β y > 0 is equivalent to γδ + γγx x γγ(x δ) x G(x ) > 0 (.15) 36

46 Recall also that 0 < δ < 1 and δ < x 1 x δ > 0. Also, since γγ > 0, we know that γγ must be sufficiently large and δ sufficiently small that Equation (.14) hold. Based on the parameter values above, we have Figure 5, which clearly satisfies Equation (.14). Figure 5: Graph of Equation (.15), that is, γδ + γγx x > 0 for δ < x 1 using δ = and γγ = MULTIPLE SOLUTIONS FOR THE COUPLED BVP USING UPPER AND LOWER SOLUTIONS METHOD Some researchers working on multiplicity of ODEs and/or PDEs have used different analytical/mathematical methods to show multiplicity to ODEs or PDEs, the work of some of them are contained in [37 39]. Some have also used experimental study to also show multiplicity [40,41] to ODEs or PDEs. Unfortunately, no one that I know has used the method of upper and lower solutions to show multiplicity. 37

47 Here, we seek to construct two pairs of upper and lower solutions for the transformed coupled system such that they divide the solution range into two disjoint regions for all [0,1]. Successful construction of these two disjoint pairs of upper and lower solutions guarantees the existence of two solutions, and consequently proves existence of multiple solutions to the BVP CONSTRUCTION OF FIRST PAIR OF UPPER AND LOWER SOLUTIONS EACH FOR yy 11 AAAAAA yy The following is a choice of upper and lower solutions pair, each for y 1 and y, y 1 = k 1 e a β1 1 y 1 = b 1 sin (π) π y = (1 δ)k e a β y = b sin(π) (.16) π where a i are constants whose limits are to be determined to satisfy the upper and lower solutions conditions (i)-(iv) [Equations (.9-.1)], 0 < b i 1 and 0 < k i < 1. This pair is chosen so that only the lower part of the solution range is covered as shown in Figures (7) and (8) is covered. As for this choice, conditions (i) and (iv) [Equations (.9) and (.1)] are clearly satisfied, as shown in section..1. We now need to prove that the other conditions, (ii) and (iii) [Equations (.10) and (.11)] are satisfied. Proof of condition (ii) [Equation (.10)]: LL 1 y 1 + F 1 (, y 1, y ) 0 LL 1 y 1 + F 1 (, y 1, y ) 38

48 The left hand side (L. H. S. ) inequality: LL 1 y 1 + F 1 (, y 1, y ) 0 L. H. S. = d y 1 d β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y = d k 1 e a 1 β 1 d β 1 k 1 e a 1 β 1 +µ 1 e β 1 1 e β 1 k 1 e a β 1 1 e β δ+e (1 δ)k e a β L. H. S. = β 1 k 1 e a 1 β 1 (1 a 1 ) +µ 1 e β 1 1 k 1 e (1 a 1 )β 1 e δ+(1 δ)k e (1 a )β For values of µ 1, we need 1 a 1 = (1 a 1 )(1 + a 1 ) 0 for the L. H. S. to be 0. An easy way to show that the L. H. S. 0 for all values of [0,1] is graphically, as shown in Figure 6. 39

49 Figure 6: Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for the pair in Equation (.16) for [0,1] for parameters β 1 = 1.5, β = 1.5, µ 1 = 0.35, µ = 0.1, δ = 0.5, γγ = and constants a 1 = a = 0.5, b 1 = b = , k 1 = 0., (1 δ)k = 0.5. Next, we prove the right hand side of condition ii in Equation (.10). The right hand side inequality: 0 LL 1 y 1 + F 1 (, y 1, y ), is equivalent to LL 1 y 1 + F 1 (, y 1, y ) 0 L. H. S. = d y 1 d β 1 y 1 +µ 1 e β 1 1 e β 1 y 1 e β δ+e y d b 1 = π sin (π) d β 1 b 1 π sin(π) +µ 1 e β 1 1 e β 1 b 1 sin (π) e π β δ+e b πsin (π) 40

50 = b β 1 π sin(π) +µ 1 e β 1 1 b 1 π eβ1 sin (π) e δ+ b β π e sin (π) This is non-negative as shown in Figure 6. Proof of condition (iii) [Equation (.11)]: LL y + F (, y 1, y ) 0 LL y + F (, y 1, y ) The left hand side inequality: LL y + F (, y 1, y ) 0 L. H. S. = d y d β y + ( µ )e β 1 e β 1 y 1 e β δ+e y = d (1 δ)k e a β d β (1 δ)e a β +( µ )e β 1 e β 1 b 1 π sin (π) e L. H. S. = β (1 δ)k e a β (1 a ) β δ+e (1 δ)k e a β +( µ )e β 1 b 1 π eβ1 sin (π) e δ+(1 δ)k e (1 a )β 41

51 This is non-positive as shown in Figure 6. The right hand side inequality: 0 LL y + F (, y 1, y ), is equivalent to LL y + F (, y 1, y ) 0 L. H. S. = d y d β y + ( µ )e β 1 e β 1 y 1 e β δ+e y d b = π sin (π) d β b π sin(π) +( µ )e β 1 e β 1 k 1 e a β 1 1 e β δ+e b π sin(π) L. H. S = b 1 + β π sin(π) +( µ )e β 1 k 1 e (1 a 1 )β 1 e δ+ b β π e sin(π) This is non-negative as shown in Figure 6. 4

52 ... CONSTRUCTION OF SECOND (AND DISJOINT) PAIR OF UPPER AND LOWER SOLUTIONS EACH FOR yy 11 AAAAAA yy Here, the pair of upper and lower solutions chosen is y 1 = 0.5e 0.5 β1 y 1 = 0.35e 0.95β β y = 0.75e y = 0.35e 0.95β (.17) This pair was chosen so that only the upper part of the solution range, as shown in Figures 8 and 9, is covered and that it is disjoint from the pair in Equation (.16). As for this choice, conditions (i) and (iv) [Equations (.9) and (.1)] are clearly satisfied as shown in section..1. We now need to prove the other conditions, (ii) and (iii) [Equations (.10) and (.11)]. Using the same parameters values as were used in constructing the upper and lower solutions in Equation (.16), we have the graphs in Figure 7, which prove conditions (ii) and (iii) [Equations (.10) and (.11)]. 43

53 Figure 7: Graphs of conditions (ii) and (iii) [Equations (.10) and (.11)] for pair in Equation (.17) for [0,1] for parameters β 1 = 1.5, β = 1.5, µ 1 = 0.35, µ = 0.1, δ = 0.5, γγ = and constants a 1 = a = 0.5, b 1 = b = Next, we show, graphically, that pair defined in Equation (.4) is disjoint from the one defined in Equation (.15) each for y 1 and y. Figure 8: Graphs that show that the pair in Equation (.16) is disjoint from the pair in Equation (.17) for y 1 for all [0,1] for parameters β = 1.5, and δ = 0.5; constants a 1 = 0.5, b 1 = , and (1 δ)k = 0.5. Graphs a and b are for the pair in Equation (.5) and c and d for the one in Equation (.4). 44

54 Figure 9: Graphs that show that the pair in Equation (.16) is disjoint from the pair in Equation (.17) for y for all [0,1] for parameters β = 1.5, and δ = 0.5; constants a = 0.5, b = , and (1 δ)k = 0.5. Graphs a and b are for the pair in Equation (.17) and c and d for the one in Equation (.16). Note: (1) In Figures 8 and 9, Graph d is not identically ero, but significantly positively close to ero. It is the sine function defined in Equation (.16) with infinitesimally small coefficient b i = () The pair in Equation (.16) is below the pair in Equation (.17), no overlapping whatsoever for all [0,1]. Remark.: Interesting thing about the Figures 8 and 9 is that we were able to construct two disjoint pairs of upper and lower solutions for all [0,1] for the transformed BVP. Since each of those pairs/regions has at least one solution [9], then, we conclude that at least two solutions exist for the BVP for the indicated choice of parameters. The region between Graphs a and b combined with the region between graphs c and d means that there exist at least two solutions to the BVP with the choice 45

55 of parameter set. Hence, this gave us the possibility of existence of multiple solutions to the BVP..3 RESULTS AND DISCUSSION The results of this work are summaried and then discussed in this section..3.1 RESULTS The nonlinearities in this BVP are caused by Arrhenius-type dependencies of the rate coefficients. This implies that for every single solution x, there is a corresponding solution x 1. This is to say that if in the original BVP, x is unique, then we can be sure that x 1 is also unique. By successfully constructing a pair [Equation (.13)] of upper and lower solutions to the coupled BVP such that Equations (.9) through (.1) are all satisfied for the set of parameters chosen, then we have that a solution (at least one) exists for the BVP for that particular set of parameters. When the Pe clet numbers and the activation energy are sufficiently large, and the Damko hler number and the reactor length are sufficiently small, a pair of upper and lower solutions was constructed for the BVP such that all the hypotheses in 46

56 Theorem.1 are satisfied, then we have that a unique solution exists for that set of parameters. However, when the Pe clet numbers and the activation energy are sufficiently small, and the Damko hler number and the reactor length are sufficiently large, at least for the parameter set chosen here, two disjoint pairs [Equations (.16) and (.17)] of upper and lower solutions were successfully constructed, and each one has a solution. This implies that at least two solutions exist for the BVP. By applying the Schauder fixed-point argument, we know that the number of solutions can only be odd, so we must have at least three solutions for the BVP in the case when the Pe clet numbers and the activation energy are small and the Damko hler number and the reactor length are large. This existence of multiple solutions is discussed in [41]..3. DISCUSSION For a uniquely predetermined/preset function x, the coupled BVP reduces to a scaler linear BVP in x 1. Thus, multiple solutions can never result. However, for a uniquely predetermined/preset function x 1, the BVP reduces to a scalar nonlinear BVP in x, with Arrhenius-type nonlinearities. Because the nonlinearities still exist in this case, multiple solutions can exist for certain sets of parameters. With large Pe clet numbers and activation energy with small Damko hler number and reactor length, the nonlinearities in the coupled BVP vanish, and the 47